Can a glass object be made to act completely transparent to only the object contained inside of it?
Yes, this is possible. 

To do this, we can make a glass material which allows some light rays through without refracting them based on certain conditions.

To get a good idea of which rays we want to let through, it's a good idea to try and visualize the possible ways rays could bounce through your scene (at least up to a few bounces):


This diagram doesn't show all possible light paths for three bounces, this is just for demonstration purposes.

I've highlighted the ray which we want to ignore in order to illuminate Suzanne as if the glass sphere wasn't there in green. Keep in mind that rays travel from the camera.

So as a solution for this particular case, you could create a node setup that ignores rays which:


Have bounced a maximum of once
Are not transmission rays
Are colliding with the inside of the glass




Result (200 samples):



For comparison, here's the same scene rendered with just a glass shader (2000 samples):